Klarinet Archive - Posting 000043.txt from 1999/01

From: Tony@-----.uk (Tony Pay)
Subj: Re: [kl] Inharmonic partials, was RE: [kl] re:Intonation training
Date: Sat, 2 Jan 1999 17:57:08 -0500

On Sat, 2 Jan 1999 09:56:45 -0600 (CST), pergler@-----.edu said:

> The important point here is that the geometry of the pipe is a
> selector of which small number of frequencies of the complex
> nonharmonic spectrum present in the driving air column is
> propagated.
>
> Now consider a complex musical instrument like a clarinet. The bore
> shape is complex, there are tone holes, which we cover in some
> pattern with our fingers. As an *approximation*, via a fudge for the
> nature of the driven oscillations by a single reed, we can
> approximate it as a perfect pipe of one of the geometries above (I
> forget right now, someone will jump in with the answer) of the
> length of the distance to the first uncovered tonehole, so that
> resonant frequencies are at odd-integer multiples of the base
> frequency.
>
> This approximation, due to great ingenuity by clarinet makers for
> compensatory adjustments, is generally very good and thus we hear a
> beautifully in-tune note at a given pitch. However, it is only an
> approximation, as we can see via forked fingerings (the geometry
> beyond the first hole does matter) and overblown fingerings (opening
> vents or holes encourages certain nodes to form and changes the
> pattern of resonant fingerings). The sheer presence of (even closed)
> tone holes actually changes the geometry. The resonance freq pattern
> is perhaps best described as "usually very close to harmonically
> related". Where it is not, we here diffuse pitch or worse.

According to Benade, whom you quote, the harmonicity isn't determined by
the close approximation of the (complex geometry) real clarinet to an
equivalent (simple geometry) tube.

Rather, the harmonicity is determined by the steady state repeating
waveform, as a consequence of Fourier's theorem plus some facts about
our perception of sound. The nonlinearity of the reed vibration at
reasonably strong dynamics guarantees the presence of higher partials.
Then, the normal modes of vibration of the tube for a given fingering
(plus the open lattice fingerholes below the fingering) usually aren't
so far away from the harmonic series appropriate to the length of tube
for that fingering, so the resonances corresponding to these normal
modes amplify the higher partials. The upshot is that the partials that
'sit' well in the peaks of these resonances are more amplified than the
ones that don't.

Because of the nonharmonicity of the modes of resonance of the
instrument (not the nonharmonicity of the resultant *sound*), the
perceived pitch in the steady state may be different from the
fundamental corresponding to the length of tube. According to Benade
(Physics of wind instrument tone and response, 1971) the fundamental
frequency locks into position in such a way as to maximise the weighted
average height of all resonance values R1, R2, R3,.... corresponding to
the harmonics f1, 2f1, 3f1,....

This is a bit arcane, but the message is that the pitch of a note on the
clarinet may depend on the dynamic, but the resultant sound for 'normal'
behaviour of the reed is *always* harmonic, though it may be more or
less rich, or indeed have a different pitch, according as the sequence
of normal modes of vibration of the clarinet are more or less harmonic.

You can hear that the sequence of normal modes of vibration of a
clarinet differ fairly appreciably from harmonicity. This is essential
to guarantee that when you open the register key, the twelfths are
sufficiently well in tune. (Overblowing without the register key,
corresponding to the normal modes, is flat at the bottom and the top of
the instrument, and true only around B flat / F.)

So paradoxically, a 'well in tune' clarinet has its resonance peaks in
the 'wrong' place for resonant sound in the lower register! There is a
trade-off.

For 'abnormal' behaviour of the reed, which I don't understand, but is
talked about in 'Fundamentals of Musical Acoustics' pp 559 -567,
multiphonics are generated by what are called heterodyne relationships
(essentially sums and differences). Benade himself says that he was
initially confused by the existence of multiphonics, which seemed to
throw his previous picture of how wind instruments operated into
question.

Of course, Benade's viewpoint may still be open to question. A
scientific friend who has worked in acoustics tells me that it may be
that some of these phenomena may be best understood in a time-domain
rather than a frequency-domain analysis, because we are able to detect
the effect of aperiodicity with greater sensitivity than the instruments
we use to perform spectral analysis.

But again, I'm simply reporting, not really understanding.

Tony
--
_________ Tony Pay
|ony:-) 79 Southmoor Rd Tony@-----.uk
| |ay Oxford OX2 6RE GMN family artist: www.gmn.com
tel/fax 01865 553339

... Life is boring without scapegoats. (I blame the scapegoats for this.)

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